KeSchn and I pointed out in the comments that this it is not possible to represent all stock dynamics using the Generalized Black Scholes model. For example, there can be jumps at random moments and not just at random moments but also jumps of random size. These jumps can affect either $\mu_t$ or $\sigma_t$. Models with too many sources of randomness are not considered useful but at least 2 extra source can be useful.
What does this say about the Martingale Representation Theorom. Doesn't that claim that stock dynamics can be captured using an Itô process? Unfortunately, the theorem is a bit more narrow (Wikipedia):
The martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian motion.
Emphasis mine. As long if there is one Brownian motion driving the randomness, all is good. The theorem doesn't hold with more sources than one.